Nonperturbative Quantum Gravity (1203.3591v1)

Abstract: Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum. "Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Ho\v{r}ava-Lifshitz gravitational models.

• The paper demonstrates that applying CDT yields three distinct phases, including an extended de Sitter-like universe emerging from causal triangulations.
• The methodology discretizes spacetime using Regge action and Monte Carlo simulations to evaluate the phase space of quantum geometries.
• The study finds a dynamic flow of the spectral dimension from about 4 at large scales to 2 at the Planck scale, suggesting a natural resolution to ultraviolet divergences.

Overview of "Nonperturbative Quantum Gravity"

The field of quantum gravity seeks a consistent and unified theory that combines quantum mechanics and general relativity, a challenge complicated by the fact that traditional perturbative approaches to scattering amplitudes in quantum gravity are nonrenormalizable. This research paper investigates a nonperturbative approach to quantum gravity using the framework of Causal Dynamical Triangulations (CDT), aiming to explore the ultraviolet (UV) properties of spacetime and their implications for a fundamental theory of quantum gravity.

CDT is based on a discretization of spacetime, where the sum over geometries in the path integral formulation of quantum gravity is restricted to causal geometries built from piecewise linear blocks. In CDT, spacetime is divided into a series of spatial slices, allowing for a discrete time evolution that preserves causal structure.

Key Components and Results

• CDT Framework: CDT distinguishes itself from Euclidean Dynamical Triangulation (DT) by maintaining a causal order in the triangulations. This is implemented via a lattice structure with discrete time steps, such that spatial slices have fixed topology and only the causal structure of spacetime is summed over. The approach takes into account the quantum analogue of a proper-time slicing in canonical quantum gravity.
• Action and Measure: Within CDT, the Regge action is used to discretize the Einstein-Hilbert action. The path integral over geometries is thus recast as the sum over all causal triangulations with weights given by the exponential of the discretized action.
• Numerical Simulations and Phases: Monte Carlo simulations play a vital role in exploring the CDT phase space. The paper identifies three distinct phases—labelled A, B, and C—based on the values of bare coupling constants:
  • Phase A exhibits highly crumpled configurations.
  • Phase B features collapsed geometries with a smaller number of large simplices.
  • Phase C corresponds to extended de Sitter-like universes.
• Emergence of Macroscopic Features: In analyzing phase C, numerical results reveal the emergence of a large-scale geometry resembling a Euclidean de Sitter space (a four-sphere), with quantum fluctuations characterized by a minisuperspace model. This suggests that CDT is capable of dynamically generating consistent macroscopic geometries.
• Spectral Dimension: The research measures the spectral dimension of the CDT universe, which dynamically changes with the scale. Remarkably, the spectral dimension transitions from a value near four at large scales to approximately two at the Planck scale. This spectral dimension flow is an intriguing feature that aligns well with certain approaches in both asymptotic safety and Hořava-Lifshitz gravity, which predict a dimensional reduction at very short distances.

Implications and Prospects

The CDT framework presents a mathematically well-defined approach to quantum gravity, showing promise in addressing the issue of nonrenormalizability by potentially being asymptotically safe. The results indicate that CDT can produce a macroscopic universe with features similar to those predicted by classical cosmology while also offering insights into the quantum nature of spacetime at Planckian scales.

In particular, the observed dimensional reduction at small scales hints at a natural resolution to ultraviolet divergences—a phenomenon that could have significant implications for the consistency and universality of quantum gravity theories. Future research may focus on further analytical and numerical studies to understand the behavior of CDT across different phases, especially near critical transitions, and to explore the possibility of incorporating matter fields seamlessly within this framework.